restricting area of UV project to projector plane
Orientation, scale and position of the projector will influence the UV project modifier.

So, starting from your configuration (I changed nothing except limiting the rendering of the plane to wire), simply scale and grab the projector.



About a 'better way' to do it. I don't know if it is a better way, but you can get rid of the UV Project modifier defining a UV map for the sphere, because the UV of the sphere will be inherited by the tile during the Boolean modifier operation.